{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,26]],"date-time":"2026-04-26T09:21:31Z","timestamp":1777195291117,"version":"3.51.4"},"reference-count":45,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2023,5,12]],"date-time":"2023-05-12T00:00:00Z","timestamp":1683849600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this work, we consider the Boiti\u2013Leon\u2013Manna\u2013Pempinelli equation with the M-truncated derivative (BLMPE-MTD). Our aim here is to obtain trigonometric, rational and hyperbolic solutions of BLMPE-MTD by employing two diverse methods, namely, He\u2019s semi-inverse method and the extended tanh function method. In addition, we generalize some previous results. As the Boiti\u2013Leon\u2013Manna\u2013Pempinelli equation is a model for an incompressible fluid, the solutions obtained may be utilized to represent a wide variety of fascinating physical phenomena. We construct a large number of 2D and 3D figures to demonstrate the impact of the M-truncated derivative on the exact solution of the BLMPE-MTD.<\/jats:p>","DOI":"10.3390\/axioms12050466","type":"journal-article","created":{"date-parts":[[2023,5,12]],"date-time":"2023-05-12T02:36:59Z","timestamp":1683859019000},"page":"466","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":21,"title":["Abundant Solitary Wave Solutions for the Boiti\u2013Leon\u2013Manna\u2013Pempinelli Equation with M-Truncated Derivative"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2394-0041","authenticated-orcid":false,"given":"Farah M.","family":"Al-Askar","sequence":"first","affiliation":[{"name":"Department of Mathematical Science, Collage of Science, Princess Nourah Bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1694-7907","authenticated-orcid":false,"given":"Clemente","family":"Cesarano","sequence":"additional","affiliation":[{"name":"Section of Mathematics, International Telematic University Uninettuno, Corso Vittorio Emanuele II, 39, 00186 Roma, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1402-7584","authenticated-orcid":false,"given":"Wael W.","family":"Mohammed","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, University of Ha\u2019il, Ha\u2019il 2440, Saudi Arabia"},{"name":"Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,5,12]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"417","DOI":"10.1016\/j.physleta.2007.07.051","article-title":"The (G\u2032\/G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics","volume":"372","author":"Wang","year":"2008","journal-title":"Phys. Lett. A"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Al-Askar, F.M., Cesarano, C., and Mohammed, W.W. (2022). The analytical solutions of stochastic-fractional Drinfel\u2019d-Sokolov-Wilson equations via (G\u2019\/G)-expansion method. Symmetry, 14.","DOI":"10.3390\/sym14102105"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Mohammed, W.W., Al-Askar, F.M., and Cesarano, C. (2022). The analytical solutions of the stochastic mKdV equation via the mapping method. Mathematics, 10.","DOI":"10.3390\/math10224212"},{"key":"ref_4","first-page":"1534067","article-title":"The Analytical Solutions of the Stochastic Fractional RKL Equation via Jacobi Elliptic Function Method","volume":"2022","author":"Mohammed","year":"2022","journal-title":"Adv. Math. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1016\/S0960-0779(02)00653-7","article-title":"Abunbant families of Jacobi elliptic function solutions of the dimensional integrable Davey-Stewartson-type equation via a new method","volume":"18","author":"Yan","year":"2003","journal-title":"Chaos Solitons Fractals"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Raheela, M., Zafar, A., Bekir, A., and Tariq, K.U. (2023). Exact wave solutions and obliqueness of truncated M-fractional Heisenberg ferromagnetic spin chain model through two analytical techniques. Waves Random Complex Media, 1\u201319.","DOI":"10.1080\/17455030.2023.2173550"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"700","DOI":"10.1016\/j.chaos.2006.03.020","article-title":"Exp-function method for nonlinear wave equations","volume":"30","author":"He","year":"2006","journal-title":"Chaos Solitons Fractals"},{"key":"ref_8","first-page":"1349","article-title":"(G\u2032\/G, 1\/G)-expansion method for traveling wave solutions of (2 + 1) dimensional generalized KdV, Sin Gordon and Landau-Ginzburg-Higgs Equations","volume":"8","author":"Iftikhar","year":"2013","journal-title":"Sci. Res. Essays"},{"key":"ref_9","first-page":"72","article-title":"The exp(-\u03d5(\u03c2))-expansion method for finding travelling wave solutions of Vakhnenko-Parkes equation","volume":"5","author":"Khan","year":"2014","journal-title":"Int. J. Dyn. Syst. Differ. Equ."},{"key":"ref_10","first-page":"1158","article-title":"New soliton solution to the longitudinal wave equation in a magneto-electro-elastic circular rod","volume":"8","author":"Seadawy","year":"2018","journal-title":"Res. Phys."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Mohammed, W.W., Alshammari, M., Cesarano, C., and El-Morshedy, M. (2022). Brownian Motion Effects on the Stabilization of Stochastic Solutions to Fractional Diffusion Equations with Polynomials. Mathematics, 10.","DOI":"10.3390\/math10091458"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"563","DOI":"10.1088\/0031-8949\/54\/6\/003","article-title":"The tanh method. I. Exact solutions of nonlinear evolution and wave equations","volume":"54","author":"Malfliet","year":"1996","journal-title":"Phys. Scr."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Al-Askar, F.M., Cesarano, C., and Mohammed, W.W. (2023). The Influence of White Noise and the Beta Derivative on the Solutions of the BBM Equation. Axioms, 12.","DOI":"10.3390\/axioms12050447"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Shah, N.A., Alyousef, H.A., El-Tantawy, S.A., Shah, R., and Chung, J.D. (2022). Analytical Investigation of Fractional-Order Korteweg\u2013De-Vries-Type Equations under Atangana\u2013Baleanu\u2013Caputo Operator: Modeling Nonlinear Waves in a Plasma and Fluid. Symmetry, 14.","DOI":"10.3390\/sym14040739"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Mohammed, W.W., Al-Askar, F.M., Cesarano, C., and Aly, E.S. (2023). The Soliton Solutions of the Stochastic Shallow Water Wave Equations in the Sense of Beta-Derivative. Mathematics, 11.","DOI":"10.3390\/math11061338"},{"key":"ref_16","first-page":"860","article-title":"New approach to a generalized fractional integral","volume":"218","author":"Katugampola","year":"2011","journal-title":"Appl. Math. Comput."},{"key":"ref_17","first-page":"1","article-title":"New approach to generalized fractional derivatives","volume":"6","author":"Katugampola","year":"2014","journal-title":"Bull. Math. Anal. Appl."},{"key":"ref_18","unstructured":"Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2016). Theory and Applications of Fractional Differential Equations, Elsevier."},{"key":"ref_19","unstructured":"Samko, S.G., Kilbas, A.A., and Marichev, O.I. (1993). Fractional Integrals and Derivatives, theory and Applications, Gordon and Breach."},{"key":"ref_20","first-page":"83","article-title":"A new truncated M fractional derivative type unifying some fractional derivative types with classical properties","volume":"16","author":"Sousa","year":"2018","journal-title":"Int. J. Anal. Appl."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"126","DOI":"10.3389\/fphy.2019.00126","article-title":"Optical Solitons With M-Truncated and Beta Derivatives in Nonlinear Optics","volume":"7","author":"Yusuf","year":"2019","journal-title":"Front. Phys."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Mohammed, W.W., El-Morshedy, M., Moumen, A., Ali, E.E., Benaissa, M., and Abouelregal, A.E. (2023). Effects of M-Truncated Derivative and Multiplicative Noise on the Exact Solutions of the Breaking Soliton Equation. Symmetry, 15.","DOI":"10.3390\/sym15020288"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Mohammed, W.W., Al-Askar, F.M., and Cesarano, C. (2022). Solutions to the (4+ 1)-Dimensional Time-Fractional Fokas Equation with M-Truncated Derivative. Mathematics, 11.","DOI":"10.3390\/math11010194"},{"key":"ref_24","first-page":"2020","article-title":"Optical solitons of fractional complex Ginzburg\u2013Landau equation with conformable, beta, and M-truncated derivatives: A comparative study","volume":"612","author":"Hussain","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"558","DOI":"10.1007\/s11082-021-03221-2","article-title":"M-truncated optical solitons to a nonlinear Schr\u00f6dinger equation describing the pulse propagation through a two-mode optical fiber","volume":"53","author":"Yusuf","year":"2021","journal-title":"Opt. Quant. Electron."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"4259","DOI":"10.1108\/HFF-10-2019-0760","article-title":"Painleve analysis for new (3 + 1)-dimensional Boiti\u2013Leon\u2013Manna\u2013Pempinelli equations with constant and time-dependent coefficients","volume":"30","author":"Wazwaz","year":"2019","journal-title":"Int. J. Numer. Methods Heat Fluid Flow"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"785","DOI":"10.1088\/0253-6102\/58\/6\/01","article-title":"Stair and step soliton solutions of the integrable (2 + 1) and (3 + 1)-dimensional Boiti\u2013Leon\u2013Manna\u2013Pempinelli equations","volume":"58","author":"Darvishi","year":"2012","journal-title":"Commun. Theor. Phys."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"103820","DOI":"10.1016\/j.rinp.2021.103820","article-title":"The exact solutions for the (3 + 1)-dimensional Boiti\u2013Leon\u2013Manna\u2013Pempinelli equation","volume":"21","author":"Duan","year":"2021","journal-title":"Results Phys."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"6277","DOI":"10.1002\/mma.5721","article-title":"A general bilinear form to generate different wave structures of solitons for a (3 + 1)-dimensional Boiti\u2013Leon\u2013Manna\u2013Pempinelli equation","volume":"42","author":"Osman","year":"2019","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"655","DOI":"10.1007\/s11071-016-3267-2","article-title":"New three-wave solutions for the (3 + 1)-dimensional Boiti\u2013Leon\u2013Manna\u2013Pempinelli equation","volume":"88","author":"Liu","year":"2017","journal-title":"Nonlinear Dyn."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"162","DOI":"10.1016\/j.aml.2017.12.011","article-title":"New non-traveling wave solutions for the (3 + 1)-dimensional Boiti\u2013Leon\u2013Manna\u2013Pempinelli equation","volume":"79","author":"Liu","year":"2018","journal-title":"Appl. Math. Lett."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"3604","DOI":"10.1016\/j.camwa.2018.02.020","article-title":"Double-periodic soliton solutions for the (3 + 1)-dimensional Boiti\u2013Leon\u2013Manna\u2013Pempinelli equation in incompressible fluid","volume":"75","author":"Liu","year":"2018","journal-title":"Comput. Math. Appl."},{"key":"ref_33","first-page":"1","article-title":"Analytical studies for the Boiti\u2013Leon\u2013Monna\u2013Pempinelli equations with variable and constant coefficients","volume":"4","author":"Pinar","year":"2019","journal-title":"Asymptot. Anal."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"715","DOI":"10.1016\/j.camwa.2018.10.008","article-title":"Breather waves and rational solutions in the (3 + 1)-dimensional Boiti\u2013Leon\u2013Manna\u2013Pempinelli equation","volume":"77","author":"Peng","year":"2019","journal-title":"Comput. Math. Appl."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"309","DOI":"10.2478\/amns.2020.1.00029","article-title":"A new approach to (3 + 1) dimensional Boiti\u2013Leon\u2013Manna\u2013Pempinelli equation","volume":"5","author":"Yel","year":"2020","journal-title":"Appl. Math. Nonlinear Sci."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1016\/j.aml.2019.05.025","article-title":"Painleve analysis, lump-kink solutions and localized excitation solutions for the (3 + 1)-dimensional Boiti\u2013Leon\u2013Manna\u2013Pempinelli equation","volume":"97","author":"Guiqiong","year":"2019","journal-title":"Appl. Math. Lett."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"234","DOI":"10.1080\/25765299.2021.1927498","article-title":"On some new soliton solutions of (3 + 1)-dimensional Boiti\u2013Leon\u2013Manna\u2013Pempinelli equation using two different methods","volume":"28","author":"Ali","year":"2021","journal-title":"Arab J. Basic Appl. Sci."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Tariq, K.U., Bekir, A., and Zubair, M. (2022). On some new travelling wave structures to the (3 + 1)-dimensional Boiti\u2013Leon\u2013Manna\u2013Pempinelli model. J. Ocean. Eng. Sci., accepted.","DOI":"10.1016\/j.joes.2022.03.015"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1007\/s11082-021-03487-6","article-title":"Complexiton and resonant multi-solitons of a (4 + 1)-dimensional Boiti\u2013Leon\u2013Manna\u2013Pempinelli equation","volume":"54","author":"Raza","year":"2022","journal-title":"Opt. Quant. Electron."},{"key":"ref_40","unstructured":"Gencyigit, M., Senol, M., and Koksal, M.E. (2023). Analytical solutions of the fractional (3 + 1)-dimensional Boiti\u2013Leon\u2013Manna\u2013Pempinelli equation. Comput. Methods Differ. Equ., 1\u201312."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1515\/TJJ.1997.14.1.23","article-title":"Semi-inverse method of establishing generalized variational principles for fluid mechanics with emphasis on turbomachinery aerodynamics","volume":"14","author":"He","year":"1997","journal-title":"Int. J. Turbo Jet-Engines"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"847","DOI":"10.1016\/S0960-0779(03)00265-0","article-title":"Variational principles for some nonlinear partial differential equations with variable coefficients","volume":"19","author":"He","year":"2004","journal-title":"Chaos Solitons Fractals"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1141","DOI":"10.1142\/S0217979206033796","article-title":"Some asymptotic methods for strongly nonlinear equations, Internat","volume":"20","author":"He","year":"2006","journal-title":"J. Mod. Phys. B"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"2420","DOI":"10.1016\/j.camwa.2009.03.026","article-title":"He\u2019s variational method for the Benjamin\u2013Bona\u2013Mahony equation and the Kawahara equation","volume":"58","author":"Ye","year":"2009","journal-title":"Comput. Math. Appl."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"1769","DOI":"10.1016\/j.apm.2015.08.018","article-title":"The modified extended tanh-function method and its applications to the Bogoyavlenskii equation","volume":"40","author":"Zahran","year":"2016","journal-title":"Appl. Math. Model."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/5\/466\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:33:29Z","timestamp":1760124809000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/5\/466"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,5,12]]},"references-count":45,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2023,5]]}},"alternative-id":["axioms12050466"],"URL":"https:\/\/doi.org\/10.3390\/axioms12050466","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,5,12]]}}}